Generalized spectra and applications to finite distributive lattices

Abstract

In our previous paper on frames of continuous functions, the classical adjunction between topological spaces and frames was generalized to a setup in which an arbitrary topological frame replaces the two element chain. The relevant composition of adjoints yields an endofunctor on topological spaces which in general fails to be idempotent. In this paper we prove a formula for iterations of this functor under certain conditions. We apply our result to the construction of finite free distributive lattices and Boolean algebras.